The level of pairs of polynomials

Given a polynomial f with coefficients in a field of prime characteristic p, it is known that there exists a differential operator that raises to its p th power. We first discuss a relation between the "level" of this differential operator and the notion of "stratification" in th...

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Bibliographic Details
Published in:Communications in algebra 2020-10, Vol.48 (10), p.4235-4248
Main Authors: Boix, Alberto F., Noordman, Marc Paul, Top, Jaap
Format: Article
Language:English
Subjects:
Differential equations
Differential operators
first order differential equation
Frobenius map
Operators (mathematics)
ordinary curve
Polynomials
Primary 13A35
prime characteristic
Secondary 13N10
supersingular curve
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Summary:Given a polynomial f with coefficients in a field of prime characteristic p, it is known that there exists a differential operator that raises to its p th power. We first discuss a relation between the "level" of this differential operator and the notion of "stratification" in the case of hyperelliptic curves. Next, we extend the notion of level to that of a pair of polynomials. We prove some basic properties and we compute this level in certain special cases. In particular, we present examples of polynomials g and f such that there is no differential operator raising g/f to its p th power.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2020.1759614